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Register a new userAnomalies of Minimal N=(0, 1) and N=(0, 2) Sigma Models on Homogeneous Spaces
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We study chiral anomalies in $mathcal N=(0, 1)$ and $(0, 2)$ two-dimensional minimal sigma models defined on generic homogeneous spaces $G/H$. Such minimal theories contain only (left) chiral fermions and in certain cases are inconsistent because of incurable anomalies. We explicitly calculate the anomalous fermionic effective action and show how to remedy it by adding a series of local counter-terms. In this procedure, we derive a local anomaly matching condition, which is demonstrated to be equivalent to the well-known global topological constraint on $p_1(G/H)$. More importantly, we show that these local counter-terms further modify and constrain curable chiral models, some of which, for example, flow to nontrivial infrared superconformal fixed point. Finally, we also observe an interesting relation between $mathcal N=(0, 1)$ and $(0, 2)$ two-dimensional minimal sigma models and supersymmetric gauge theories. This paper generalizes and extends the results of our previous publication arXiv:1510.04324.
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The two-dimensional minimal supersymmetric sigma models with homogeneous target spaces $G/H$ and chiral fermions of the same chirality are revisited. We demonstrate that the Moore-Nelson consistency condition revealing a global anomaly in CP(N-1) (with N>2 and ${mathcal N}=(0,2)$ supersymmetry) due to a nontrivial first Pontryagin class is in one-to-one correspondence with the local anomalies of isometries in these models. These latter anomalies are generated by fermion loop diagrams which we explicitly calculate. In the case of O}(N) sigma models the first Pontryagin class vanishes, so there is no global obstruction for the minimal ${mathcal N}=(0,1)$ supersymmetrization of these models. We show that at the local level isometries in these models are anomaly free. Thus, there are no obstructions to quantizing the minimal ${mathcal N}=(0,1)$ models with the $S^{N-1}= SO(N)/SO(N-1)$ target space. This also includes CP(1) (equivalent to $S^{2}$) which is an exceptional case from the CP(N-1) series. We also discuss a relation between the geometric and gauged formulations of the CP}(N-1) models.
We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation. The anomalous dimensions are calculated up to two loops implying that one of the beta functions is explicitly known up to three loops. The fixed point in the ratio of the couplings found previously at one loop is not shifted at two loops. We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the Konishi anomaly. This gives us another method for finding exact $beta$ functions. We prove that despite the chiral nature of the models under consideration quantum loops preserve isometries of the target space.
A similarity transformation, which brings a particular class of the $N=1$ string to the $N=0$ one, is explicitly constructed. It enables us to give a simple proof for the argument recently proposed by Berkovits and Vafa. The $N=1$ BRST operator is turned into the direct sum of the corresponding $N=0$ BRST operator and that for an additional topological sector. As a result, the physical spectrum of these $N=1$ vacua is shown to be isomorphic to the tensor product of the $N=0$ spectrum and the topological sector which consists of only the vacuum. This transformation manifestly keeps the operator algebra.
In this paper we apply the previously derived formalism of permutation orbifold conformal field theories to N=2 supersymmetric minimal models. By interchanging extensions and permutations of the factors we find a very interesting structure relating various conformal field theories that seems not to be known in literature. Moreover, unexpected exceptional simple currents arise in the extended permuted models, coming from off-diagonal fields. In a few situations they admit fixed points that must be resolved. We determine the complete CFT data with all fixed point resolution matrices for all simple currents of all Z_2-permutations orbifolds of all minimal N=2 models with k eq 2 mod 4.
Following recent work on GLSM localization, we work out curvature couplings for rigidly supersymmetric nonlinear sigma models with superpotential for general target spaces, describing both ordinary and twisted chiral superfields on round two-sphere worldsheets. We briefly discuss why, unlike four-dimensional theories, there are no constraints on Kahler forms in these theories. We also briefly discuss general issues in topological twists of such theories.
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